Theoretical Analysis of Vernier-Effect-Induced Sensitivity Enhancement of Dual Fiber Fabry-Pérot Cavities in OFDRs

Abstract

The vernier-effect-based sensitivity enhancement of two kinds of sensing units consisting of dual fiber Fabry-Pérot (FP) cavities in the Optical Frequency Domain Reflectometry (OFDR) is analyzed in this paper. Theoretical analysis reveals that significant differences exist in the sensitivity enhancement between the cascaded and parallel dual fiber FP cavties when demodulated by an OFDR system. When the conditions of the vernier effect are satisfied, the sensing unit with cascaded FP cavities does not exhibit a sensitivity enhancement compared to a single FP sensor, whereas the sensing unit with parallel FP cavities can achieve an enhanced sensitivity. This phenomenon differs from that observed in direct wavelength interrogation systems. The results are further verified with numerical simulations on the temperature sensing. When the vernier-effect conditions are met, the sensitivity of the sensing unit with cascaded FP sensors is 9.99 pm/°C, while the sensitivity of the sensing unit with parallel FP sensors can reach up to 128.97 pm/°C. The findings of this paper provide valuable insights for the design of high-sensitive distributed optical fiber sensing systems.

1. Introduction

Optical fiber sensors have been widely applied in various fields, such as civil engineering, energy monitoring, biomedicine, and aerospace [1,2,3,4] due to their compact sizes, high sensitivities, immunities to electromagnetic interference and compatibilities to harsh environments. Among them, fiber-optic Fabry-Pérot (FP) sensors play an important role for their high sensitivities and simple structures. They have been employed in the measurement of temperature, strain, pressure, refractive index, acoustic wave and so on [5,6,7,8,9,10,11]. Moreover, singular optical phenomena arising from light–matter interactions, such as polarization transformations at reflection points, also underline the broader significance of FP-based fiber sensors [12].

The performance of a single FP sensor is limited by its relatively low sensitivity. Although a higher sensitivity can be achieved by optimizing the structure, the sensitivity remains restricted if only a single FP cavity is employed without additional enhancement mechanisms. To address this problem, researchers have proposed vernier-effect-based sensors with dual fiber FP cavities, including the cascaded FP cavities and the parallel FP cavities [13,14,15]. The vernier effect amplifies the sensitivity by constructing two FP cavities with slightly different optical path differences (OPDs). The superimposed envelope is more sensitive to minor changes in the cavity length or the refractive index of the sensing FP cavity. However, the research on vernier-effect-based highly sensitive sensors based on dual fiber FP cavities is still limited to the single-point sensing systems [16,17,18,19,20], where the direct wavelength interrogation system is used.

In many application scenarios, the spatial distribution of the physical states is also very important and a distributed sensing system is desirable [21,22,23,24]. The Optical Frequency Domain Reflectometry (OFDR) has become a powerful tool for the high-density distributed sensing [25,26,27,28,29,30]. By combining the OFDR system and FP sensing units, a precise localization and an effective interrogation of the physical states can be obtained simultaneously. In 2014, Ou et al. combined FP sensors with an OFDR system to conduct strain sensing experiments, and 26 sensors can be interrogated accurately at the same time [31]. In 2021, Zhu et al. employed FP sensors and an OFDR system for the refractive index sensing, and obtained a sensitivity comparable to that of conventional FP sensors [32]. Liu et al. adopted Pd-coated FP sensors as high-sensitivity hydrogen sensing units and realized a spatially distributed hydrogen leakage detection with a coverage area up to the kilometer scale in [33]. In 2024, Feng et al. connected two FP sensors in series via a single optical fiber, with one sensor for temperature measurement and the other for humidity measurement, and the simultaneous temperature and humidity monitoring was achieved in an OFDR system [34]. However, in these studies, FP cavities are considered independently. How to construct the vernier-effect-based dual fiber FP cavities in OFDR systems remains to be investigated.

In this paper, the vernier-effect-based sensitivity enhancement of two types of dual fiber FP cavities in OFDR systems is analyzed. It is revealed that the behavior of the dual fiber FP cavities demodulated by the OFDR systems is quite different from that demodulated by the direct wavelength interrogation systems. For the cascaded dual fiber FP cavities, there are two working modes under which the vernier effect conditions are satisfied, denoted by the mode I and mode II. In both modes, the envelope induced by the Vernier effect is sensitive to variations in the OPDs, but no sensitivity enhancement is observed. For the parallel dual fiber FP cavities, when the vernier effect conditions are met, a significant sensitivity enhancement can be achieved. The simulation results indicate that, under the Vernier effect condition, the temperature sensitivity of the cascaded FP sensor unit is 9.99 pm/°C. The parallel dual fiber FP cavities can achieve a temperature sensitivity up to 128.97 pm/°C, approximately increased by 12.86 times compared to a single FP sensor. The investigation results of this paper provide valuable insights for the design of high-sensitive distributed optical fiber sensing systems.

2. Theoretical Analysis

2.1. Dual Fabry-Pérot Cavities Demodulated by OFDR Systems

Figure 1a illustrates the basic structure of a multiplexed sensing system based on the OFDR. The system consists of a tunable laser source (TLS), a photodetector (PD), a fiber Mach-Zehnder (MZ) interferometer, and multiple sensing units. Light from the TLS is divided into two beams by a coupler. One goes through the reference arm, and the other through the sensing arm. Light reflected from different sensing units along the sensing arm interferes with the light through the reference arm by another coupler, and the signal is detected by the PD. Owing to the different time delays introduced by each sensing unit, the positions of the sensing units can be obtained by analyzing the corresponding beat frequencies.

The demodulation process of the system is illustrated in Figure 1b–d. The light reflected by the i-th sensing unit exhibits a time delays $\tau$ relative to the reference light, as shown in Figure 1b. The data acquisition card (DAQ) captures a composite signal composed of the superposition of multiple beat frequency signals. By performing a Fast Fourier Transform (FFT) on the mixed signal, the individual beat frequencies corresponding to each sensing unit can be distinguished in the frequency domain. The frequency-domain data can be converted into the spatial (distance) domain, thereby enabling the localization of each sensing unit, as shown in Figure 1c. Then, by applying an Inverse Fast Fourier Transform (IFFT), the peaks of interest are extracted from the distance domain and transformed into the wavelength domain, as shown in Figure 1d. The physical state variations measured by each sensing unit can be demodulated. As shown in Figure 1e, the physical model of the FP sensor used in the simulation is constructed by introducing two reflective surfaces (e.g., two micro-holes) on a single-mode fiber, forming an FP sensor [35]. The cavity in between is made of SMF with a refractive index of 1.4682. To ensure that at least one full spectral period can be observed within the selected wavelength range (1530–1570 nm), a cavity length of 371 μm was chosen.

Assuming the incident light is a linearly frequency-swept light with a sweep rate of $\gamma$ Hz/s. The output electric field of the reference arm can be expressed as:

$$E_{ref}={\sqrt{a}E}_{0}exp\left\{j\left[2\pi f_{0}t+\pi \gamma t^2+\varphi \left(t\right)\right]\right\}$$

(1)

where $E_0$ is the amplitude of the field, $f_0$ is the initial frequency, and $\varphi \left(t\right)$ accounts for the random phase fluctuations in the light source. The parameter $a$ denotes the power splitting ratio from the coupler to the reference arm, $\gamma$ is frequency tuning rate of the TLS (tunable laser source), and the time t represents an arbitrary instant within the duration during which the wavelength sweeps from 1530 nm to 1570 nm at the specified tuning rate. The light reflected back from the sensing unit can be expressed as:

$$E_{sen}={\sqrt{\left(1-a\right)r}E}_{0}exp\left\{j\left[2\pi f_0(t-\tau )+\pi \gamma ({t-\tau )}^2+\varphi (t-\tau )\right]\right\}$$

(2)

where $r$ is the reflectivity of the sensor, and $\tau$ is the related time delay. The time delay can be obtained by $\tau =2nl/c$, and $l$ represents the position of the sensing unit. The optical intensity detected by the PD can be expressed as:

$$\begin{array}{c}I=\left(E_{ref}+E_{sen}\right){\left(E_{ref}+E_{sen}\right)}^{*}\\ =\left[a+r(1-a)\right]{E_0}^2+2\sqrt{ar\left(1-a\right)}cos(2\pi \gamma \tau t+2\pi f_0\tau -\pi \gamma {\tau }^2)\end{array}$$

(3)

where $f_b=\gamma \tau$ is the beat frequency.

Assuming the wavelength sweep range of the TLS is from 1530 to 1570 nm, and the sweep rate is approximately 80 nm/s [32]. Three FP sensors are placed at positions of 4 m, 5 m, and 6 m along the sensing arm, which has 99% of the power of the incident light. The three FPs all have the same refractive index and cavity length. The simulation results are shown in Figure 2.

The simulated spectrum of the system is shown in Figure 2a. In the time domain, it is difficult to distinguish the spectral contributions from individual sensing units. By applying the FFT to the time-domain waveform, the signal is converted into the spatial (distance) domain, from which the positions of different sensing units can be identified. As shown in Figure 2b, three distinct peaks appear at the positions of 4 m, 5 m, and 6 m, corresponding to the three FP sensors. Additional peaks at around 1 m and 2 m are attributed to the beat frequencies generated between the reflections of different FP sensors. Applying Hanning windows to extract the reflection peaks at different positions in the distance domain. The enlarged signal at 4 m is as shown in Figure 2c. After zero-padding and performing the IFFT, the reflection spectra of the three FP sensors can be obtained, as illustrated in Figure 2d.

The optical vernier effect can be utilized to enhance the sensitivity of sensors. Vernier-effect-based dual FP cavities can be categorized into two configurations: cascaded and parallel structures. One FP cavity serves as the sensing cavity, whose refractive index and cavity length are $n_s$ and $L_s$, respectively. The other is the reference cavity, whose refractive index and cavity length are $n_r$ and $L_r$, respectively. In traditional wavelength-demodulation systems, both configurations can effectively enhance the sensing sensitivity when $n_s{L}_s\approx n_r{L}_r$. The sensitivity amplification factor $G$ of the vernier effect is:

$$G={\displaystyle \frac{n_s{L}_s}{n_s{L}_s-n_r{L}_r}}$$

(4)

Similarly, the cascaded and parallel dual fiber FP cavities can be employed in the OFDR system, as shown in Figure 3. $R_i$ and ${\alpha }_i$ denote the reflectivity and transmissivity of the reflective surfaces $M_i$, respectively. $n_i$ and $L_i$ represent the refractive index and cavity length of each FP cavity. In the cascaded structure, three reflective ends $M_i$ form three FP cavities: FP₁, FP₂ and the hybrid cavity FP₁ + FP₂. In the parallel structure, FP₁ is formed by $M_1$ and $M_2$; and FP₂ is formed by $M_3$ and $M_4$. When demodulated by the OFDR, it remains to be investigated whether the sensitivities can both be enhanced with these two types of dual fiber FP cavities with the vernier effect.

2.2. The Analysis of the Vernier Effect in the OFDR System

2.2.1. Cascaded Dual FP Cavities

As shown in Figure 3a, the structure of cascaded dual FP cavities has three reflective surfaces. In a direct wavelength interrogation system, it is called a three-beam interferometer. If the field of the incident light is $E_{in}$, and the intensities of the three reflected beams $I_1$, $I_2$, and $I_3$ can be wrote as:

$$\left\{\begin{array}{c}{I}_1=R_1{\left|E_{in}\right|}^2\\ I_2={\left(1-{\alpha }_1\right)}^2{\left(1-R_1\right)}^2{R}_2{\left|E_{in}\right|}^2\\ { I}_3={(1-{\alpha }_1)}^2{(1-R_1)}^2{(1-{\alpha }_2)}^2{(1-R_2)}^2{R}_3{\left|E_{in}\right|}^2\end{array}\right.$$

(5)

The total reflected electric field is given by:

$$\begin{array}{c} E_r=E_{in}\left[\sqrt{R_1}+\left(1-{\alpha }_1\right)\left(1-R_1\right)\sqrt{R_2}{e}^{-j{\varphi }_1}+\right.\\ \left.\left(1-{\alpha }_1\right)\left(1-R_1\right)\left(1-{\alpha }_2\right)\left(1-R_2\right)\sqrt{R_3}{e}^{-j{2(\varphi }_1+{\varphi }_2)}\right]\end{array}$$

(6)

where ${\varphi }_1=2\pi n_1{L}_1/\lambda$ is introduced by FP₁, ${\varphi }_2=2\pi n_2{L}_2/\lambda$ is introduced by FP₂, ${\varphi }_1+{\varphi }_2$ is introduced by the hybrid cavity FP₁ + FP_2, and $\varphi$ represents the phase difference induced by the propagation of light within the Fabry-Pérot (FP) cavity. The total reflected intensity can be expressed as:

$$I_r=I_1+I_2+I_3+2\sqrt{I_1{I}_2}\mathit{cos}\left(2{\varphi }_1\right)+2\sqrt{I_2{I}_3}\mathit{cos}\left(2{\varphi }_2\right)+2\sqrt{I_1{I}_3}\mathit{cos}\left[2({\varphi }_1+{\varphi }_2)\right]$$

(7)

In a vernier-effect-based three-beam interferometer described by Equation (7), one of the two individual cavities is typically chosen as the sensing cavity while the other serves as the reference cavity. However, the hybrid cavity will introduce a crosstalk, which should be suppressed in the spectrum before demodulation in a direct wavelength interrogation system.

In an OFDR system, the demodulated signal introduced by the dual fiber FP cavities is not only determined by the phase delays, but also by the beat frequencies. Let the time delay between the light reflected by the sensing unit and that through the reference arm is ${\tau }_A({\tau }_A={\displaystyle \frac{2nl}{c}})$, the time delay induced by the FP₁ is ${\tau }_1({\tau }_1={\displaystyle \frac{2n_1{L}_1}{c}})$, and the delay induced by the FP₂ is ${\tau }_2({\tau }_2={\displaystyle \frac{2n_2{L}_2}{c}})$. The electrical field reflected by the front-end of the FP₁ can be obtained based on Equation (2) as:

$$E_1=\sqrt{\left(1-a\right)r}\sqrt{R_1}exp\left\{j\left[2\pi f_0(t-{\tau }_A)+\pi \gamma ({t-{\tau }_A)}^2\right]\right\}$$

(8)

The electrical field reflected by the rear ends of the FP₁ and FP₂ is:

$$\begin{array}{c}{E}_2=\sqrt{\left(1-a\right)r}\left(1-{\alpha }_1\right)\left(1-R_1\right)\sqrt{R_2}\\ \ast exp\left\{j\left[2\pi f_0(t-{{\tau }_A-\tau }_1)+\pi \gamma ({t-{{\tau }_A-\tau }_1)}^2\right]\right\}\\ +\sqrt{\left(1-a\right)r}\left(1-{\alpha }_1\right)\left(1-R_1\right)\left(1-{\alpha }_2\right)\left(1-R_2\right)\sqrt{R_3}\\ \ast exp\left\{j\left[2\pi f_0(t-{{\tau }_A-\tau }_1-{\tau }_2)+\pi \gamma ({t-{{\tau }_A-\tau }_1-{\tau }_2)}^2\right]\right\}\end{array}$$

(9)

Thus, the total reflected field is $E_{sen}=E_1+E_2$. Substituting $E_{sen}$ into Equations (1) and (3), the interference light intensity can be expressed as:

$$\begin{array}{c}{I_r=A}^2+B^2+T^2+P^2+2ABcos\left(2\pi \gamma {\tau }_{A}t-\pi \gamma {{\tau }_A}^2+M{\tau }_A\right)\\ +2ATcos\left[2\pi \gamma \left({\tau }_A+{\tau }_1\right)t-\pi \gamma {\left({\tau }_A+{\tau }_1\right)}^2+M\left({\tau }_A+{\tau }_1\right)\right]\\ +2APcos\left[2\pi \gamma \left({\tau }_A+{\tau }_1+{\tau }_2\right)t-\pi \gamma {\left({\tau }_A+{\tau }_1+{\tau }_2\right)}^2+M\left({\tau }_A+{\tau }_1+{\tau }_2\right)\right]\\ +2BTcos\left[2\pi \gamma {\tau }_{1}t-\pi \gamma {\tau }_1\left({\tau }_1+2{\tau }_A\right)+M{\tau }_1\right]\\ +2BPcos\left[2\pi \gamma {(\tau }_1+{\tau }_2)t-\pi \gamma {\left({\tau }_A+{\tau }_1+{\tau }_2\right)}^2-\pi \gamma {{\tau }_A}^2+M({\tau }_1+{\tau }_2)\right]\\ +2PTcos\left[2\pi \gamma {\tau }_{2}t-\pi \gamma {\left({\tau }_A+{\tau }_1+{\tau }_2\right)}^2-\pi \gamma ({{\tau }_A+{\tau }_1)}^2+M{\tau }_2\right]\end{array}$$

(10)

where the expressions of A, B, M, T, and P are as follows:

$$\left\{ \begin{array}{c}A=\sqrt{a}\\ B=\sqrt{\left(1-a\right)r}\sqrt{R_1}\\ M=2\pi f_0\\ T=\sqrt{\left(1-a\right)r}\left(1-{\alpha }_1\right)\left(1-R_1\right)\sqrt{R_2}\\ P=\sqrt{\left(1-a\right)r}\left(1-{\alpha }_1\right)\left(1-R_1\right)\left(1-{\alpha }_2\right)\left(1-R_2\right)\sqrt{R_3}\end{array}\right.$$

(11)

As shown in Equation (10), the detected optical intensity contains six frequency components, corresponding to six peaks in the spatial domain. However, a Hanning window is applied prior to the inverse fast Fourier transform (IFFT) to isolate the reflection peaks near the position of the sensing unit in OFDR systems. The window width is typically a few times the length of the cavity, which is in the order of micrometers. As a result, only the frequency components near the one corresponding to the physical position are of interest in an OFDR system. Because the time delay ${\tau }_A$ is much larger than ${\tau }_1$ and ${\tau }_2$, the frequency components of $\gamma {\tau }_A$, $\gamma ({\tau }_A+{\tau }_1)$ and $\gamma ({\tau }_A+{\tau }_1+{\tau }_2)$ can be extracted. The other three frequency components in Equation (10) correspond to spatial positions near 0 m, and will be excluded. Among the three frequency components being considered, $\gamma {\tau }_A$ is used to determine the location of the sensor, while $\gamma ({\tau }_A+{\tau }_1)$ and $\gamma ({\tau }_A+{\tau }_1+{\tau }_2)$ are used for sensing. This implies that the working dual FP cavities in the OFDR system is the FP₁ and the hybrid cavity FP₁ + FP₂. This is different from the vernier-effect-based cascaded dual FP cavities in the direct wavelength demodulation system, where FP₁ and FP₂ serve as the sensing and reference cavities, respectively. The high-frequency component generated by the hybrid cavity can be utilized for sensing in an OFDR system, and the crosstalk does not exist any longer. When the vernier effect is considered for the cascaded dual fiber FP cavities in an OFDR system, two operating modes can be employed. In mode I, the FP₁ is the sensing cavity, and the FP₁ + FP₂ is the reference cavity. In this case, $n_s{L}_s=n_1{L}_1$, $n_r{L}_r=n_1{L}_1+n_2{L}_2$. In mode II, the FP₁ + FP₂ is the sensing cavity, and the FP₁ is the reference cavity. In this case, $n_s{L}_s=n_1{L}_1+n_2{L}_2$, $n_r{L}_r=n_1{L}_1$.

In mode I, the vernier effect condition is $n_1{L}_1\approx n_1{L}_1+n_2{L}_2$ (FP₁ is the sensing cavity, and FP₁ + FP₂ is the reference cavity.). The peak wavelength of the envelope can be expressed as:

$${\lambda }_e={\displaystyle \frac{2\left(n_s{L}_s-n_r{L}_r\right)}{m}}={\displaystyle \frac{2n_2{L}_2}{m}}$$

(12)

where m denotes the interference order of the fringe. When the FP cavity is subjected to external environmental changes, like a temperature variation of $∆T$, the cavity length $L$ and the cavity refractive index $n$ will change due to the thermal expansion effect and thermo-optic effect. The wavelength shift $∆{\lambda }_e$ can be obtained as:

$${\displaystyle \frac{{\mathrm{\Delta }\lambda }_e}{\mathrm{\Delta }T}}={\lambda }_e({\displaystyle \frac{1}{n_2}}{\displaystyle \frac{\mathrm{\Delta }{n}_2}{\mathrm{\Delta }T}}+{\displaystyle \frac{1}{L_2}}{\displaystyle \frac{\mathrm{\Delta }{L}_2}{\mathrm{\Delta }T}})$$

(13)

Here, it is assumed that only FP₁ is subjected to external temperature modulation, while FP₂ remains unaffected. As can be seen from Equation (13), the temperature sensitivity depends only on FP₂. However, according to the previous assumption, FP₂ is not affected by temperature in this case. Therefore, the interference envelope does not experience a wavelength shift with temperature changes. Consequently, the Vernier-effect-based envelope neither enables effective sensing nor enhances sensitivity.

If both FP₁ and FP₂ are subjected to temperature modulation, the relationship between the envelope shift in the wavelength domain and temperature can still be described by Equation (13). It can be seen that although FP₁ is also affected by temperature, the wavelength shift of the final envelope depends only on the cavity length and refractive index of FP₂. Even if temperature variations induce changes in the refractive index or cavity length of FP₁, as long as the Vernier condition is satisfied, no spectral distortion will occur, and the observed wavelength shift of the envelope remains unaffected. In other words, the envelope shift is determined solely by FP₂. Since a cascaded FP sensor operates as a sensing unit in temperature measurement, the entire unit will inevitably be influenced by temperature. Therefore, in this mode, both FP₁ and FP₂ are affected by temperature. Although temperature sensing can still be achieved under such conditions, the envelope shift is governed only by FP₂, and thus the sensitivity enhancement effect of the Vernier principle cannot be realized.

In mode II, the vernier effect condition is $n_1{L}_1+n_2{L}_2\approx n_1{L}_1$ (FP₁ + FP₂ is the sensing cavity, and FP₁ is the reference cavity.). In this case, both FP₁ and FP₂ are affected by temperature. Substituting $n_s{L}_s=n_1{L}_1+n_2{L}_2$ and $n_r{L}_r=n_1{L}_1$ into Equation (12), the relationship between the wavelength shift of the interference envelope and temperature variation still satisfies Equation (13). In this case, the wavelength shift of the envelope in the wavelength domain still depends solely on FP₂ and is independent of FP₁. Temperature sensing can still be performed in this mode; however, since the envelope shift is determined only by FP₂, the sensitivity enhancement provided by the Vernier effect cannot be realized.

In summary, for the cascaded FP sensors demodulated by OFDR analyzed in this work, mode I and mode II are equivalent. As a sensing unit, they can perform temperature measurements normally; however, the Vernier-based sensitivity enhancement cannot be achieved, and the operation is effectively equivalent to using only FP₂ for sensing.

However, when the FP₂ is used alone as a sensing unit, the wavelength shift $∆{\lambda }_s$ can be expressed as:

$${\displaystyle \frac{{\mathrm{\Delta }\lambda }_s}{\mathrm{\Delta }T}}={\lambda }_s({\displaystyle \frac{1}{n_2}}{\displaystyle \frac{{\mathrm{\Delta }n}_2}{\mathrm{\Delta }T}}+{\displaystyle \frac{1}{L_2}}{\displaystyle \frac{{\mathrm{\Delta }L}_2}{\mathrm{\Delta }T}})$$

(14)

Comparing Equation (14) with Equation (13), it is evident that the sensitivity of the cascaded FP sensor is identical to that of the second FP sensor used independently. No sensitivity enhancement due to the vernier effect can be obtained. This will be further validated through simulations in the next section.

According to Equation (7), it can be observed that in the direct wavelength demodulation system, temperature sensing is realized through ${\varphi }_1$ and ${\varphi }_2$, thereby achieving the Vernier effect. Either FP₁ or FP₂ can be employed as the sensing cavity, while the other serves as the reference cavity. Assuming FP₁ is the sensing cavity and FP₂ is the reference cavity, the Vernier condition is satisfied when $n_s{L}_s=n_1{L}_1$, $n_r{L}_r=n_1{L}_1$. Accordingly, the wavelength shift $∆{\lambda }_e$ induced by a temperature change $∆T$ can be expressed as

$$\begin{array}{c}{\displaystyle \frac{{\mathrm{\Delta }\lambda }_e}{\mathrm{\Delta }T}}={\displaystyle \frac{{\lambda }_e}{n_1{L}_1-n_2{L}_2}}\left(L_1{\displaystyle \frac{\mathrm{\Delta }{n}_1}{\mathrm{\Delta }T}}+n_1{\displaystyle \frac{\mathrm{\Delta }{L}_1}{\mathrm{\Delta }T}}-L_2{\displaystyle \frac{\mathrm{\Delta }{n}_2}{\mathrm{\Delta }T}}-n_2{\displaystyle \frac{\mathrm{\Delta }{L}_2}{\mathrm{\Delta }T}}\right)\\ ={\lambda }_{e}G({\displaystyle \frac{1}{n_1}}{\displaystyle \frac{\mathrm{\Delta }{n}_1}{\mathrm{\Delta }T}}+{\displaystyle \frac{1}{L_1}}{\displaystyle \frac{\mathrm{\Delta }{L}_1}{\mathrm{\Delta }T}})\end{array}$$

(15)

where G is the magnification factor induced by the vernier effect:

$$G={\displaystyle \frac{n_1{L}_1}{n_1{L}_1-n_2{L}_2}}$$

(16)

From the above expression, it can be seen that the final temperature sensitivity is related to both the refractive indices and cavity lengths of the two Fabry-Pérot (FP) cavities. In contrast, for the cascaded FP sensor system based on OFDR demodulation, due to the window setting in the demodulation process—or, more fundamentally, due to the intrinsic mechanism of signal demodulation in OFDR—the sensing quantities are effectively ${\varphi }_1$ and ${\varphi }_1+{\varphi }_2$, that is, $\gamma ({\tau }_A+{\tau }_1)$ and $\gamma ({\tau }_A+{\tau }_1+{\tau }_2)$. Based on Equations (13) and (14), when the Vernier effect condition is satisfied, the temperature sensitivity of the cascaded FP sensor depends only on the refractive index and cavity length of one of the two cavities, and is independent of the other (FP₂ in this analysis). Therefore, compared with the cascaded FP sensor in a direct wavelength-demodulation system, the cascaded FP sensor demodulated by OFDR analyzed in this work does not exhibit an enhancement in sensitivity.

To achieve sensitivity amplification in this structure, sensing should not be performed using $\gamma ({\tau }_A+{\tau }_1)$ and $\gamma ({\tau }_A+{\tau }_1+{\tau }_2)$; instead, it should be based on $\gamma ({\tau }_A+{\tau }_1)$ and $\gamma ({\tau }_A+{\tau }_2)$. In other words, the component ${\gamma \tau }_2$ near zero frequency (as shown in Equation 10) needs to be shifted to $\gamma ({\tau }_A+{\tau }_2)$ through software processing. This work will be carried out in future studies.

Let the thermal expansion coefficient and thermo-optic coefficient of the FP cavities in the above analysis be ${\alpha }_L$ and ${\beta }_n$, respectively. According to the thermo-optic effect at temperature T,

$$n\left(T\right)=n_0+{\beta }_{n}T$$

(17)

where $n_0$ denotes the initial refractive index of the cavity. According to the thermo-optic effect:

$$L\left(T\right)=L_0(1+{\alpha }_{L}T)$$

(18)

where $L_0$ denotes the initial cavity length. Then, Equation (13) can also be expressed as:

$${\displaystyle \frac{{\mathrm{\Delta }\lambda }_e}{\mathrm{\Delta }T}}={\displaystyle \frac{{\lambda }_e}{n_2}}({\beta }_n+n_2{\alpha }_L)$$

(19)

Thus, the temperature sensitivity of the analyzed cascaded FP sensor can be represented by the thermo-optic coefficient and thermal expansion coefficient of the the FP₂ cavity. If ${\beta }_n+n_2{\alpha }_L>0$, the envelope curve exhibits a redshift in the wavelength domain with increasing temperature; otherwise, it exhibits a blueshift.

2.2.2. Parallel Dual FP Cavities

The sensing unit with the parallel dual fiber FP cavities is illustrated in Figure 3b. Since the two sensors are structurally independent, no crosstalk issue exists in this configuration. The reflected electrical field from FP₁ is:

$$\begin{array}{c}{E}_1=\sqrt{\left(1-a\right)r}\sqrt{R_1}exp\left\{j\left[2\pi f_0(t-{\tau }_A)+\pi \gamma ({t-{\tau }_A)}^2\right]\right\}+\sqrt{\left(1-a\right)r}\left(1-{\alpha }_1\right)\\ \ast \left(1-R_1\right)\sqrt{R_2}exp\left\{j\left[2\pi f_0(t-{\tau }_A-{\tau }_1)+\pi \gamma ({t-{\tau }_A-{\tau }_1)}^2\right]\right\}\end{array}$$

(20)

and the reflected field from FP₂ is:

$$\begin{array}{c}{E}_2=\sqrt{\left(1-a\right)r}\sqrt{R_3}exp\left\{j\left[2\pi f_0(t-{\tau }_A)+\pi \gamma ({t-{\tau }_A)}^2\right]\right\}+\sqrt{\left(1-a\right)r}\left(1-{\alpha }_3\right)\\ \ast \left(1-R_3\right)\sqrt{R_4}exp\left\{j\left[2\pi f_0(t-{\tau }_A-{\tau }_2)+\pi \gamma ({t-{\tau }_A-{\tau }_2)}^2\right]\right\}\end{array}$$

(21)

The total reflected optical field can be expressed as $E_{sen}=E_1+E_2$. Correspondingly, the total light intensity can be expressed as:

$$\begin{array}{c}{I}_r=A^2+B^2+T^2+O^2+U^2+2BO+2A\left(B+O\right)cos\left[2\pi \gamma {\tau }_{A}t-\pi \gamma {{\tau }_A}^2+M{\tau }_A\right]\\ +2ATcos\left[2\pi \gamma \left({\tau }_A+{\tau }_1\right)t-\pi \gamma {\left({\tau }_A+{\tau }_1\right)}^2+M\left({\tau }_A+{\tau }_1\right)\right]\\ +2AUcos\left[2\pi \gamma \left({\tau }_A+{\tau }_2\right)t-\pi \gamma {\left({\tau }_A+{\tau }_2\right)}^2+M\left({\tau }_A+{\tau }_2\right)\right]\\ +2UTcos\left[2\pi \gamma \left({\tau }_1-{\tau }_2\right)t+\pi \gamma {\left({\tau }_A+{\tau }_2\right)}^2-\pi \gamma {\left({\tau }_A+{\tau }_1\right)}^2+M\left({\tau }_1-{\tau }_2\right)\right]\\ +2T\left(B+0\right)cos\left[2\pi \gamma {\tau }_{1}t-\pi \gamma {\left({\tau }_A+{\tau }_1\right)}^2-\pi \gamma {{\tau }_A}^2+M{\tau }_1\right]\\ +2U\left(B+0\right)cos\left[2\pi \gamma {\tau }_{2}t-\pi \gamma {\left({\tau }_A+{\tau }_2\right)}^2+\pi \gamma {{\tau }_A}^2+M{\tau }_2\right]\end{array}$$

(22)

where the expressions of O and U can be expressed as:

$$\left\{\begin{array}{c} O=\sqrt{\left(1-a\right)r}\sqrt{R_3}\\ U=\sqrt{\left(1-a\right)r}\left(1-{\alpha }_3\right)\left(1-R_3\right)\sqrt{R_4}\end{array}\right.$$

(23)

Similarly, three distinct frequency components will be extracted in the demodulation process of an OFDR. They are $\gamma {\tau }_A$, $\gamma ({\tau }_A+{\tau }_1)$ and $\gamma \left({\tau }_A+{\tau }_2\right).$ Among them, $\gamma {\tau }_A$ serves to determine the sensor’s position, while $\gamma ({\tau }_A+{\tau }_1)$ and $\gamma \left({\tau }_A+{\tau }_2\right)$ are employed for sensing. To achieve sensitivity enhancement via the vernier effect, one of the FP cavities can be employed as the sensing cavity and the other as the reference cavity. When the vernier effect condition $n_1{L}_1\approx n_2{L}_2$ is met, the temperature sensitivity of sensing unit with the parallel dual FP cavities is obtained (FP₁ as the sensing cavity):

$${\displaystyle \frac{{\mathrm{\Delta }\lambda }_e}{\mathrm{\Delta }T}}={\lambda }_{e}G({\displaystyle \frac{1}{n_1}}{\displaystyle \frac{\mathrm{\Delta }{n}_1}{\mathrm{\Delta }T}}+{\displaystyle \frac{1}{L_1}}{\displaystyle \frac{\mathrm{\Delta }{L}_1}{\mathrm{\Delta }T}})$$

(24)

where G can be obtained from Equation (16).

Comparing Equation (24) with Equation (14), the vernier effect can be realized with the parallel dual FP cavities when the OFDR is used for the demodulation.

3. Simulation Results

3.1. Cascaded Dual FP Cavities Demodulated by the OFDR

3.1.1. OFDR Demodulation Systems

Table 1. Basic parameters of the sensor.

Sensing Unit		n	L (μm)
Cascaded dual FP cavities	FP1	1	1000
	FP2	1.4682	61
Parallel dual FP cavities	FP1	1.4682	371
	FP2	1	500
Single FP cavity	FP	1.4682	371

presents the detailed parameters of all the FP sensors used in this work. The cascaded dual fiber FP cavities are employed as the sensing units and integrated into the OFDR system, as shown in Figure 4a. Compared with the structure mentioned earlier, an auxiliary interferometer is added. This interferometer is typically based on a Mach–Zehnder configuration, consisting of two branches with different lengths. Its purpose is to obtain the real-time phase of the laser source, thereby compensating for the broadening caused by nonlinear phase noise. The light source also provides a trigger signal to the DAQ card to ensure synchronization between the laser and the DAQ system. A polarization controller (PC) is placed in the reference arm of the main interferometer to reduce polarization effects. Three sensing units with cascaded FP cavities are placed at positions of 4 m, 5 m, and 6 m along the sensing arm. The three sensing units are configured to meet the vernier effect as analyzed in Section 2. Figure 4b illustrates the physical model of the cascaded FP sensor structure, which is composed of SMF-HCF-SMF [13]. The first FP cavity is formed by the HCF with a refractive index of 1, while the second cavity is made of SMF with a refractive index of 1.4682. To satisfy the Vernier effect condition ($n_1{L}_1\approx n_1{L}_1+n_2{L}_2$) and to ensure that at least one full spectral period can be observed within the selected wavelength range (1530–1570 nm), the cavity lengths were chosen as $L_1=1000 \mathrm{\mu }\mathrm{m}$ and $L_2=61 \mathrm{\mu }\mathrm{m}$ after careful consideration. The wavelength scanning range of the TLS is set to 1530–1570 nm, with a tuning speed of 80 nm/s.

The simulation results are as shown in Figure 5. The time-domain plot of the original beat signal reveals that the signal contains numerous spikes, as shown in Figure 5a. This irregularity arises from the addition of noise during simulation. By performing an FFT on the original beat signal, the power spectrum can be obtained. The frequency domain signal is then converted into the distance domain signal, as shown in Figure 5b. Distinct reflection peaks can be observed at the positions of 4 m, 5 m, and 6 m, which are consistent with the predefined positions of the sensing units. The mutual beat frequencies between sensors can also be observed. The peaks at 4 m, 5 m and 6 m are magnified for observation, as shown in Figure 5c–e. It can be seen that three distinct peaks exist at each position, corresponding to the frequency components of $\gamma {\tau }_A$, $\gamma ({\tau }_A+{\tau }_1)$ and $\gamma ({\tau }_A+{\tau }_1+{\tau }_2)$, respectively. The simulation results verify that, when combining cascaded dual fiber FP cavities with an OFDR system, only the first cavity directly connected to the optical path FP₁ and the hybrid cavity FP₁ + FP₂ can be employed to realize the vernier effect. This is consistent with the analysis presented in Section 2.2.1.

3.1.2. Vernier Effect Verification

According to Equation (13), when temperature acts on the cascaded FP sensor shown in Figure 4b, the wavelength shift of the interference envelope depends only on FP₂, that is, on the optical path difference (OPD) of FP₂, for which OPD = 2$n_2{L}_2$. Since FP₂ is a quartz cavity, both its thermal expansion coefficient and thermo-optic coefficient are positive. Substituting Equations (17) and (18) into OPD = 2$n_2{L}_2$, it can be seen that the OPD of FP₂ increases with rising temperature.

Since FP₁ is an air cavity enclosed by a single-mode fiber (SMF), its cavity length is primarily affected by the thermal expansion of the SMF. The thermal expansion coefficient of the SMF (quartz fiber) is denoted as ${\alpha }_{SMF}=5.5\times {10}^{-7}/°\mathrm{C}$, the thermo-optic coefficient of the air cavity as ${\beta }_{HCF}=-9.0\times {10}^{-7}/°\mathrm{C}$, and that of the SMF as ${\beta }_{SMF}=1.0\times {10}^{-5}/°\mathrm{C}$. Substituting these values into Equations (17) and (18) yields the temperature-dependent relationships for the cavity lengths and refractive indices of each FP cavity in the cascaded sensor. As analyzed in Section 2, mode I and mode II are equivalent. Here, the hybrid cavity is assumed as the sensing cavity, and FP₁ as the reference cavity. The sensitivities of the cascaded FP sensor and the individual FP₂ sensor to temperature variations were simulated, with the temperature ranging from 10 °C to 90 °C in 10 °C steps. The simulations consider a sensing unit located 4 m along the fiber. Figure 6a shows the spectrum of the cascaded FP sensor at an environmental temperature of 10 °C. As expected from the previous design, the demodulated spectrum exhibits a typical Vernier envelope. The X-axis represents wavelength, and the Y-axis denotes normalized intensity. Figure 6b,c show the locally magnified spectra of the cascaded FP sensor when both FP₁ and FP₂ vary with temperature, and when only FP₂ varies with temperature, respectively. Figure 6d shows the locally magnified spectrum of the FP₂ sensor alone under temperature variations. Figure 6e depicts the relationship between the envelope shift in the wavelength domain and temperature when both FP₁ and FP₂ in the cascaded sensor vary with temperature, indicating a temperature sensitivity of 9.99 pm/°C. Figure 6f shows the wavelength-domain envelope shift versus temperature for the cascaded sensor when only FP₂ varies with temperature, and for FP₂ alone as a sensing unit. Comparison indicates that both cases exhibit nearly identical sensitivity and similar linear correlation coefficients. Comparing Figure 6b,c,e,f, it can be seen that for the OFDR-demodulated cascaded FP sensor, the envelope shift in the wavelength domain with temperature is the same whether both FP₁ and FP₂ vary with temperature or only FP₂ varies. The temperature sensitivity and linear correlation are identical, confirming that the cascaded sensor’s temperature response is determined solely by FP₂, independent of FP₁. This simulation validates the analysis presented in Section 2. Figure 6f further shows that the temperature sensitivity of the cascaded FP sensor as a sensing unit is equivalent to that of the FP₂ sensor alone.

In summary, the cascaded FP sensor combined with OFDR can perform temperature sensing normally but does not achieve sensitivity enhancement, consistent with the analysis in Section 2. Compared with a single FP sensor, the cascaded structure is more complex and does not provide increased sensitivity. Whether this configuration offers improvements in measurement stability, such as reduced errors, remains a subject for future investigation.

3.2. Parallel Dual FP Cavities Demodulated by the OFDR

3.2.1. OFDR Demodulation Systems

Figure 7a illustrates the system configuration of the parallel dual FP cavities as a sensing unit integrated with OFDR for temperature sensing. Three sensing units are placed at the positions of 4 m, 5 m, and 6 m, respectively. Each sensing unit contains two FP cavities, one serving as a reference cavity and the other as a sensing cavity. Figure 7b shows the physical model of the parallel FP sensor structure. FP₁ is the same as that shown in Figure 1e [35], while FP₂ is formed by splicing SMFs on both sides of an HCF [7], with the HCF section in the middle serving as the FP sensor, having a refractive index of 1. To satisfy the Vernier effect condition ($n_1{L}_1\approx n_2{L}_2$) and to ensure that at least one full spectral period can be observed within the wavelength range of 1530–1570 nm, the cavity lengths were chosen as $L_1=371 \mathrm{\mu }\mathrm{m}$ and $L_2=500 \mathrm{\mu }\mathrm{m}$ after careful consideration. The FP₁ is selected as the sensing cavity.

Figure 8a shows the time-domain signal of the original beat frequency signal. By applying a FFT to the beat signal, the power spectrum is obtained. Converting the frequency domain into the spatial domain yields the position information shown in Figure 8b. It can be observed that there are distinct peaks at 4 m, 5 m, and 6 m. The three peaks of the distance domain signal are further magnified for a detailed observation, as shown in Figure 8c–e. It is evident that three peaks can be clearly seen in Figure 8d,e, while only two are visually discernible in Figure 8c. According to the previous analysis, the three peaks in the spatial domain correspond to the frequency components of $\gamma {\tau }_A$, $\gamma ({\tau }_A+{\tau }_1)$ and $\gamma \left({\tau }_A+{\tau }_2\right)$, respectively. Since the optical path lengths of the two cavities are designed to be approximately equal in order to achieve the vernier effect, the corresponding peaks in the spatial domain are located very close to each other or may even overlap. The frequency component $\gamma {\tau }_A$ is used for localization, and the frequency components $\gamma ({\tau }_A+{\tau }_1)$ and $\gamma \left({\tau }_A+{\tau }_2\right)$ are used for sensing purposes.

3.2.2. Vernier Effect Verification

Similarly, two configurations of the sensing unit are compared, one is a single FP sensor and the other is a sensing unit composed of parallel dual FP cavities. The ambient temperatures range from 10 °C to 90 °C, with an increment of 10 °C. Figure 9a shows the simulated spectrum obtained from the dual FP cavities, from which a typical vernier-effect envelope is observed. Figure 9b presents the enlarged envelops under different temperatures, and Figure 9c presents the enlarged spectrum obtained from the single FP cavity under different temperatures. Figure 9d illustrates the comparison of the two configurations of the linear fitting results of the relationships between the peak wavelengths and the temperature. The comparison reveals that the parallel dual FP cavities, used as the sensing unit, exhibit a temperature sensitivity of 128.97 pm/°C, with a linear correlation coefficient of 0.9999. The single FP cavity has a temperature sensitivity of 10.03 pm/°C, and a linear correlation coefficient of 0.9999. The sensing unit with the parallel dual FP cavities achieves a 12.86-fold enhancement in temperature sensitivity compared to a single FP cavity. This indicates that, when combined with OFDR, only the parallel dual FP cavities can realize the vernier effect and achieve an enhanced sensitivity.

4. Discussion

Since this work is primarily based on theoretical analysis and lacks comparison with experimental results, part of the theoretical analysis is combined with the experimental results reported in [34]. In [34], the authors fabricated two types of FP cavities: one similar to the single-air-single structure shown in Figure 7b, where a hollow-core fiber is employed as the FP cavity, and another where a solidified PVA solution is used as the FP cavity. The temperature sensitivities of both structures were provided; however, for the structure using solidified PVA solution as the FP cavity, the refractive index parameter was not reported. Therefore, in this section, the structure with a hollow-core fiber as the FP cavity, as described in [34], is adopted to verify part of the theoretical analysis presented in this work. The detailed parameters of this structure are listed in

Table 2. Parameters of the FP sensor [34].

Structure	Sensing Type	Cavity Length (μm)	Refractive Index	Spatial Positioning (m)	Sensitivity
SMF-HCF-SMF	Temperature sensing	151.67	1	7.7; 10.043; 12.899	3.87 pm/°C

.

The comparison results are shown in Figure 10. Figure 10a presents the spatial positioning information of the sensors. It can be observed that there are distinct peaks at approximately 7.7 m, 10 m, and 12.9 m, corresponding to the locations of the three FP sensors, which is consistent with the results reported in [34]. Figure 10b shows a magnified view of the sensor at 7.7 m, where two peaks are visible, corresponding to the two reflecting surfaces of the FP sensor. Subsequently, the temperature sensitivity of the FP sensor was simulated for a temperature change from 25 °C to 50 °C with a step of 5 °C. Figure 10c presents the locally magnified spectrum obtained by applying an IFFT-based demodulation to the data in Figure 10a. Figure 10d shows the relationship between the spectral wavelength shift of the FP sensor and the temperature change. A temperature sensitivity of 3 pm/°C is observed, which agrees well with the results reported in [34].

Since the analysis in this work is conducted under ideal conditions, the temperature gradient between the reference cavity and the sensing cavity in the parallel FP sensors demodulated by OFDR may affect the maintenance of the Vernier condition in practical engineering scenarios. In the present simulation setup, the reference cavity is assumed to remain constant, and thus the variation in the reference cavity does not need to be considered when measuring temperature changes. However, in practice, if the reference cavity is subjected to temperature modulation, the measured temperature variations may deviate from the actual values, as shown in Figure 11. Figure 11a presents the spectrum when the reference cavity remains unaffected (kept at 10 °C), and the sensing cavity environment is also 10 °C. Figure 11b illustrates the case where the reference cavity environment is heated to 100 °C while the sensing cavity remains at 10 °C. It can be observed that although the reference cavity experiences significant environmental disturbance, the observed spectrum still maintains a Vernier envelope. However, compared with the case where both cavities are at 10 °C, the spectral envelope exhibits a shift in the wavelength domain. Figure 11c shows the simulated results when the reference cavity environment is maintained at 10 °C, increased to 20 °C, or decreased to 0 °C, while the sensing cavity temperature increases from 10 °C to 90 °C with a step of 10 °C. Under the condition of the reference cavity environment rising to 20 °C, each 1 °C increase in the sensing cavity temperature results in an additional wavelength shift of 3.53 pm (132.5–128.97 pm), i.e., about 2.7% larger than the normal case (reference cavity at 10 °C). Similarly, when the reference cavity decreases to 0 °C, the deviation becomes approximately 2.3% smaller. Figure 11d illustrates the simulated results for reference cavity environments of 10 °C, 15 °C, and 5 °C, respectively, with the sensing cavity temperature varying from 10 °C to 90 °C. It can be seen that in this case, the three scenarios yield almost identical sensitivities, indicating that small variations in the reference cavity temperature may not affect the final measurement results.

Therefore, in practical applications, significant environmental fluctuations on the reference cavity may not hinder the realization of the Vernier effect but could lead to deviations in the measured temperature variations (the deviation being positive or negative depending on the reference cavity condition). To mitigate this, the reference cavity in long-distance distributed sensing systems can be placed within controlled cabinets at network nodes (e.g., base stations or aggregation points) rather than directly deployed in exposed outdoor segments, ensuring its temperature stability within ±0–5 °C. In addition, a software-based compensation approach can be adopted, in which a reference channel is introduced into the OFDR data stream to monitor the real-time phase drift of the reference cavity and digitally subtract the drift from the sensing channel results (i.e., digital de-drifting).

5. Conclusions

In summary, the vernier-effect-based sensitivity enhancement of two kinds of sensing units consisting of dual fiber FP cavities in the OFDR is analyzed in this paper. To the knowledge of the authors, the conditions required to achieve the vernier effect in cascaded and parallel dual FP cavities demodulated by the OFDR for multi-point sensing are theoretically discussed for the first time. When the conditions of the vernier effect are satisfied, the sensing unit with cascaded FP cavities does not exhibit a sensitivity enhancement compared to a single FP sensor, whereas the sensing unit with parallel FP cavities can achieve an enhanced sensitivity. Numerical simulations are carried out for both configurations of dual fiber FP cavities. The results show that the cascaded dual FP cavities can achieve a temperature sensitivity of 9.99 pm/°C. For the parallel dual FP cavities, the temperature sensitivity can reach 128.97 pm/°C when the vernier effect conditions are met, which is approximately 12.86 times higher than that of a single FP sensor. The investigations of this paper provide valuable insights for the design of high-sensitive distributed optical fiber sensing systems.